Need help to remove a table lamp socket

My brand new lamp doesn't come with a harp base (saddle). To use my vintage
shade (spider fitter), I bought a harp with a saddle and began disassembling my
lamp for the installation. The socket top cover and inner socket were easily
removed but the socket base is stuck. There's no set screw so I'm assuming I
would just have to unscrew it, but the nut below keeps spinning.
Without removing the socket base, the space for the nut is too tight for my
wrench. Any ideas on how to remove the socket base?
BTW, I also tried pulling upward...in case it isn't a screw on...
Thanks in advance!

Thanks Andy,
That's ok. I ended up pulling off the felt base and unscrewing the bottom nut to
give me enough slack to unscrew the bottom of the socket. Didn't want to
disassemble the lamp this much just to install the saddle, but the problem is
now solved.

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